Divide using synthetic division $$ ( 2.5x^{3}+6x^{2}-5.5x-10 ) \div ( x+1 ) $$
The synthetic division table is:
$$ \begin{array}{c|rrrr}-1&\dfrac{ 5 }{ 2 }&6&-\dfrac{ 11 }{ 2 }&-10\\& & -\dfrac{ 5 }{ 2 }& -\dfrac{ 7 }{ 2 }& \color{black}{9} \\ \hline &\color{blue}{\dfrac{ 5 }{ 2 }}&\color{blue}{\dfrac{ 7 }{ 2 }}&\color{blue}{-9}&\color{orangered}{-1} \end{array} $$The solution is:
$$ \dfrac{ \dfrac{ 5 }{ 2 }x^{3}+6x^{2}-\dfrac{ 11 }{ 2 }x-10 }{ x+1 } = \color{blue}{\dfrac{ 5 }{ 2 }x^{2}+\dfrac{ 7 }{ 2 }x-9} \color{red}{~-~} \dfrac{ \color{red}{ 1 } }{ x+1 } $$Step 1 : Write down the coefficients of the dividend into division table. Put the zero from $ x + 1 = 0 $ ( $ x = \color{blue}{ -1 } $ ) at the left.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&\frac{ 5 }{ 2 }&6&-\frac{ 11 }{ 2 }&-10\\& & & & \\ \hline &&&& \end{array} $$Step 1 : Bring down the leading coefficient to the bottom row.
$$ \begin{array}{c|rrrr}-1&\color{orangered}{ \frac{ 5 }{ 2 } }&6&-\frac{ 11 }{ 2 }&-10\\& & & & \\ \hline &\color{orangered}{\frac{ 5 }{ 2 }}&&& \end{array} $$Step 2 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \frac{ 5 }{ 2 } } = \color{blue}{ -\frac{ 5 }{ 2 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&\frac{ 5 }{ 2 }&6&-\frac{ 11 }{ 2 }&-10\\& & \color{blue}{-\frac{ 5 }{ 2 }} & & \\ \hline &\color{blue}{\frac{ 5 }{ 2 }}&&& \end{array} $$Step 3 : Add down last column: $ \color{orangered}{ 6 } + \color{orangered}{ \left( -\frac{ 5 }{ 2 } \right) } = \color{orangered}{ \frac{ 7 }{ 2 } } $
$$ \begin{array}{c|rrrr}-1&\frac{ 5 }{ 2 }&\color{orangered}{ 6 }&-\frac{ 11 }{ 2 }&-10\\& & \color{orangered}{-\frac{ 5 }{ 2 }} & & \\ \hline &\frac{ 5 }{ 2 }&\color{orangered}{\frac{ 7 }{ 2 }}&& \end{array} $$Step 4 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \frac{ 7 }{ 2 } } = \color{blue}{ -\frac{ 7 }{ 2 } } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&\frac{ 5 }{ 2 }&6&-\frac{ 11 }{ 2 }&-10\\& & -\frac{ 5 }{ 2 }& \color{blue}{-\frac{ 7 }{ 2 }} & \\ \hline &\frac{ 5 }{ 2 }&\color{blue}{\frac{ 7 }{ 2 }}&& \end{array} $$Step 5 : Add down last column: $ \color{orangered}{ -\frac{ 11 }{ 2 } } + \color{orangered}{ \left( -\frac{ 7 }{ 2 } \right) } = \color{orangered}{ -9 } $
$$ \begin{array}{c|rrrr}-1&\frac{ 5 }{ 2 }&6&\color{orangered}{ -\frac{ 11 }{ 2 } }&-10\\& & -\frac{ 5 }{ 2 }& \color{orangered}{-\frac{ 7 }{ 2 }} & \\ \hline &\frac{ 5 }{ 2 }&\frac{ 7 }{ 2 }&\color{orangered}{-9}& \end{array} $$Step 6 : Multiply by the number on the left, and carry the result into the next column: $ \color{blue}{ -1 } \cdot \color{blue}{ \left( -9 \right) } = \color{blue}{ 9 } $.
$$ \begin{array}{c|rrrr}\color{blue}{-1}&\frac{ 5 }{ 2 }&6&-\frac{ 11 }{ 2 }&-10\\& & -\frac{ 5 }{ 2 }& -\frac{ 7 }{ 2 }& \color{blue}{9} \\ \hline &\frac{ 5 }{ 2 }&\frac{ 7 }{ 2 }&\color{blue}{-9}& \end{array} $$Step 7 : Add down last column: $ \color{orangered}{ -10 } + \color{orangered}{ 9 } = \color{orangered}{ -1 } $
$$ \begin{array}{c|rrrr}-1&\frac{ 5 }{ 2 }&6&-\frac{ 11 }{ 2 }&\color{orangered}{ -10 }\\& & -\frac{ 5 }{ 2 }& -\frac{ 7 }{ 2 }& \color{orangered}{9} \\ \hline &\color{blue}{\frac{ 5 }{ 2 }}&\color{blue}{\frac{ 7 }{ 2 }}&\color{blue}{-9}&\color{orangered}{-1} \end{array} $$Bottom line represents the quotient $ \color{blue}{ \frac{ 5 }{ 2 }x^{2}+\frac{ 7 }{ 2 }x-9 } $ with a remainder of $ \color{red}{ -1 } $.